[RPG] Does the “Downtime: Catching Up” option require DM permission, or just logging it

Question 1) You can make the checks when he goes back, or he can make the checks himself every day. The business works even if the owner isnt there to activelly direct the workers, as it is assumed you hired people to make your business work, and your earnings are your profit after you paid them and the business paid all it's cost. Though, if the hero isnt there, you should apply the 7 days absence penalties as normal, which could ruin the business after a while (which can be fixed with a manager).

Question 2) The manager's job is to prevent the absence penalties and roll for the random events instead of the hero, but the earning checks will happen wether or not the hero is there. Personally, i prefer to roll them after they get back, we count the days away, i roll a couple of random events, see how many checks they have to make, and they take 10 on all of them.

Question 3) As said above, the checks will happen regardless of his presence. But that is a good question because if the hero can communicate at distance, the absence clock is reset everytime he communicates with them. That way they know what to do and they know their boss is still alive.

Question 4) Everything other than GP you will have to pay the half-price for it. If i recall correctly, labor and goods are 20 gp (half price is 10 gp), while influence is 30 gp (half price is 15 gp), and magic is 100 gp (half price is 50 gp). So, when you are earning capital you dont gain any GP, but you are not actually losing any either, because the price for rooms and hirelings are based on the full cost of the capital, so if you take your time to earn the capital, you can buy things at half the cost. See that as an investment of your money, in the long run you will have discounts to build other rooms and hire more people.

I have been in a similar situation and found it best to just handwave away the disparity and pretend people were there all along. Give them commensurate XP and gear (None of the fancy gear) and let them pick up from where the others left off.

If you try to give the missing players extra attention to explain how they join up what you are inadvertently doing is punishing the players who turned up both weeks.

As an improvement to this advice:

Since you know it is going to happen you might prefer to run your campaign in a series of short stories, each session ending back in town where the group simply comes together based on whatever quests are on offer at the time.

This means you can't run really complex campaigns, but completely gets around the random nature of your attendance.